Asymptotic Safety in Generalized Proca Theories

TL;DR

The paper investigates whether Generalized Proca Theories can be UV completed within a quantum-field-theory framework by applying the Functional Renormalization Group to a truncation with up to two derivatives and four powers of the Proca field. It uncovers a triplet of non-Gaussian fixed points (the Proca triplet) that remain near-equal in coupling space, among which the main Proca fixed point has a non-tachyonic mass $m^2(k)\propto g_2^* k^2>0$ and five relevant directions, suggesting a potential UV completion though with slow convergence and sensitivity to truncation. The Reuter fixed point persists in simple truncations but the Proca fixed points deviate in stability and exhibit singular behavior tied to the broken $U(1)$ symmetry and the physical longitudinal mode, indicating caution about extrapolating to the full theory space. The work highlights the need for extended truncations and further checks (e.g., positivity bounds) to establish a robust UV completion for GPTs and motivates exploring UV structures of broader massive vector–tensor theories.

Abstract

Generalized Proca Theories are the most general higher-derivative extensions of a massive vector field that retain second-order equations of motion. They are phenomenologically interesting as models of dynamical dark energy that, unlike scalar-tensor theories, can naturally accommodate cosmological anisotropies. A key open question is whether such theories can be fundamental. As a first step in this direction, we investigate whether they admit an ultraviolet completion within a quantum field theory framework, working with a truncation comprising up to four powers of the Proca field and up to two derivatives. We find a triplet of non-Gaussian ultraviolet fixed points, that lie very close to one another. Only one of them features a non-tachyonic Proca mass and could thus serve as a consistent ultraviolet completion for Generalized Proca Theories. We name it the Proca fixed point. We discuss its stability and contrast its features with those of the standard Reuter fixed point of the asymptotic safety scenario for quantum gravity and matter. In particular, we show that the Gaussian and Reuter fixed points lie on singular hypersurfaces of the flow of Generalized Proca Theories, yet can act as quasi-fixed points in certain regimes.

Figures (6)

• Figure 1: The figure shows how the fixed-point structure of GPTs changes upon introducing the first non-trivial interaction term. Turning $g_{4,1}$ on causes the Reuter fixed point (red dot) to split into three fixed points (blue dots), one of which is a pair of complex-conjugate fixed points (empty blue dot) at the full-$A^2$ order. This is revealed by tracking the evolution of the triplet as $g_{4,1}$ is gradually switched on. The connecting lines follow the zeros of the first three beta functions until the fourth one, $\beta_{g_{4,1}}$, also vanishes for specific values of $g_{4,1}^\ast$. For $g_{4,1}=0$, the Proca and gravitational fields interact minimally and the relevant fixed point is the Reuter one (red dot). Increasing $g_{4,1}$ to positive values, all four beta functions vanish at $\rm{Re}(g_{4,1}) \simeq 0.032$, yielding a complex-conjugate pair (empty blue dot). For negative $g_{4,1}$, the partial fixed point eventually splits into two real fixed points, reached at $g_{4,1}^\ast \simeq -0.072$ and $g_{4,1}^\ast \simeq -0.58$ (blue dots). No fixed-point collision occurs, since $\{\beta_\lambda,\beta_g\}$ depend linearly on $g_{4,1}$, while $\beta_{g_2}$ is cubic in it.
• Figure 2: Evolution of the fixed-point values for the dimensionless couplings of the main (circles, connected by solid lines), identical-twin (squares, connected by dashed lines), and accidental-twin (rhombuses, connected by vertically dashed lines) Proca fixed points as a function of the truncation order. At each step, we add one coupling to control the evolution and stability of the Proca fixed point and its twins; we start from the full-$A^2$ truncation, since this is the lowest order truncation where the Proca fixed point and its twins appear. Filled circles, squares, and rhombuses correspond to real fixed points, whereas empty ones denote pairs of complex-conjugate fixed points. The main Proca fixed point emerges from a pair of complex-conjugate fixed points, but it is the only one with a non-tachyonic mass.
• Figure 3: Merger of two complex-conjugate fixed points into a real one as the coupling $g_{4}$ is slowly activated. Data are produced by varying $g_{4}$ from zero to the fixed-point value $g_{4}^\ast\approx -0.24$, and solving the system $\beta_\lambda=\beta_g=\beta_{g_2}=\beta_{g_{4,1}}=0$ for each intermediate $g_4$. The colorful lines show how the imaginary parts of the fixed-point values of the other couplings diminish to eventually vanish when the collision happen. At the same point, also the beta function $\beta_{g_4}$ vanishes. Through this mechanism, the main Proca fixed point turns from a complex-conjugate pair into a real fixed point upon inclusion of the interaction coupling $g_4$.
• Figure 4: Projection of the full flow onto the $\{\lambda,g,g_2\}$ sub-theory space. The projected main and identical-twin Proca fixed points (red dots) have complex-conjugate critical exponents, which cause the characteristic spiraling behaviors of close-by RG trajectories. The two NGFPs are connected by a separatrix line.
• Figure 5: Projection of the RG flow on the $\{\lambda,g_{4,1}\}$ plane (left panel), $\{\lambda,g\}$ plane (central panel) and $\{g_2,g\}$ plane (right panel). The main, identical-twin, and accidental-twin Proca fixed points are represented by a green circle, an orange square, and a purple rhombus, respectively. In the left and right panels, the red line depicts a line of sGFP---an infinite set of sGFPs, characterized by vanishing couplings, except for $g_{4,1}$ which can take any value. Each sGFP makes the beta functions divergent due to a vanishing Proca mass, but at the same time, such sGFP can act as quasi-fixed points for some RG trajectories. In the central panel, the dark red curve depicts a singular line, where the beta functions for $g$ and $\lambda$ diverge.